Artificial Intelligence

Transcription

1 Artificial Intelligence Constraint Satisfaction Vibhav Gogate The University of Texas at Dallas Some material courtesy of Rina Dechter, Alex Ihler and Stuart Russell

2 Constraint Satisfaction Problems The constraint network model Variables, domains, constraints, constraint graph, solutions Examples: graph-coloring, 8-queen, cryptarithmetic, crossword puzzles, vision problems,scheduling, design The search space and naive backtracking, The constraint graph Maybe: Consistency enforcing algorithms arc-consistency, AC-1,AC-3 Next time: Backtracking strategies Forward-checking, dynamic variable orderings Special case: solving tree problems Local search for CSPs 2

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4 Constraint Satisfaction Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: A B, A D, D E, etc. 4

5 Constraint Satisfaction Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: A B, A D, D E, etc. A red red green green yellow yellow B green yellow red yellow green red 5

6 Constraint Satisfaction Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: A B, A D, D E, etc. A red red green green yellow yellow B green yellow red yellow green red A B C D G E F 6

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10 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 10

11 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 11

12 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 12

13 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 13

14 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 14

15 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 15

16 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 16

17 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 17

18 Sudoku Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 18

19 Sudoku Constraint propagation Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 19

20 Sudoku Constraint propagation Variables: 81 slots Domains = {1,2,3,4,5,6,7,8,9} Constraints: 27 not-equal Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints 20

21 A network of binary constraints Variables X 1,,X n X 1 Domains D 1,,D n sets of discrete values, X 5 X 2 Binary constraints R ij X 4 X 3 represent the list of allowed pairs of values; R ij µ D i D j Constraint graph: A node for each variable and an arc for each constraint Solution: An assignment of a value to each variable such that no constraint is violated. A network of constraints represents the relation of all solutions. 22

22 Varieties of constraints Unary constraints involve a single variable, e.g., SA green Binary constraints involve pairs of variables, e.g., SA WA Higher-order constraints involve 3 or more variables, e.g., cryptarithmetic column constraints 23

23 Cryptarithmetic Each letter represents a different digit They should satisfy the addition constraint

24 Cryptarithmetic Each letter represents a different digit They should satisfy the addition constraint

25 Examples Cryptarithmetic: SEND+MORE = MONEY n-ueens Crossword puzzles Graph coloring Vision problems Scheduling Assignment (who teaches what); timetable (where & when) Transportation scheduling, factory assembly, etc.

26 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are 4 ( 2 ) = 6 constraints involved: X 1 31

27 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are ( 2 ) R 12 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} = 6 constraints involved: X 1 32

28 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are ( 2 ) R 12 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} = 6 constraints involved: R 1 13 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} X 33

29 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are ( 2 ) R 12 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 13 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 14 {(1,2)(1,3)(2,1)(2,3)(2,4)(3,1)(3,2)(3,4)(4,2)(4,3)} R 23 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 24 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 34 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} = 6 constraints involved: X 1 34

30 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are ( 2 ) R 12 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 13 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 14 {(1,2)(1,3)(2,1)(2,3)(2,4)(3,1)(3,2)(3,4)(4,2)(4,3)} R 23 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 24 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 34 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} = 6 constraints involved: Constraint Graph : X 1 35

31 Example 1: The 4-queen problem Place 4 ueens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal. Domains: D i {1,2,3,4 }. Standard CSP formulation of the problem: Variables: each row is a variable. X 1 X 2 X 3 X Constraints: There are ( 2 ) R 12 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 13 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 14 {(1,2)(1,3)(2,1)(2,3)(2,4)(3,1)(3,2)(3,4)(4,2)(4,3)} R 23 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} R 24 {(1,2)(1,4)(2,1)(2,3)(3,2)(3,4)(4,1)(4,3)} R 34 {(1,3)(1,4)(2,4)(3,1)(4,1)(4,2)} = 6 constraints involved: Constraint Graph : X 1 X 3 X 2 X 4 36

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34 The search space 39

35 The search space Definition: given an ordering of the variables a state: is an assignment to a subset of variables that is consistent. Operators: add an assignment to the next variable that does not violate any constraint. Goal state: a consistent assignment to all the variables. X 1,...,X n 40

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41 Dependence on variable ordering 46

42 Recap Constraint Satisfaction problem (e.g., map coloring) Variables: (regions of a map) Domains: values that the variables can take (colors) Constraints: Restrictions on values that can be assigned simultaneously Backtracking search a state is an assignment to a subset of variables that is consistent. Operators: add an assignment to the next variable that does not violate any constraint. DFS in this state-space Select an unassigned variable and assign a value to it!!

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44 Which variable to choose next?

45 Which variable to choose next?

46 In What order should a variable s values be tried?

47 Can we detect failures early?

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51 Also called consistency enforcement techniques

52 Consistency enforcement Consistency enforcement techniques Arc-consistency (Waltz, 1972) Path-consistency (Montanari 1974, Mackworth 1977) I-consistency (Freuder 1982) Transform the network into smaller and smaller networks. 57

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58 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 63

59 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 64

60 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 65

61 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 66

62 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 67

63 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T Z = 68

64 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X T Y Z = 69

65 Arc-consistency 1 X, Y, Z, T 3 X Y Y = Z T Z X T X T Y Incorporated into backtracking search Z = 70

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75 Propositional Satisfiability Example: party problem If Alex goes, then Becky goes: A! B (or, : A Ç B) If Chris goes, then Alex goes: C! A (or, : C Ç A ) uery: Is it possible that Chris goes to the party but Becky does not? is the proposition = { : A Ç B, : C Ç A, : B, C} satsifiable? 80

76 Unit Propagation Arc-consistency for cnfs. Involve a single clause and a single literal Example: (A, : B, C) ^ (B)! (A,C) 81

77 Look-ahead for SAT (Davis-Putnam, Logeman and Laveland, 1962) 82

78 Look-ahead for SAT: DPLL example: (~A B) (~C A) (A B D) (C) (Davis-Putnam, Logeman and Laveland, 1962) Backtracking look-ahead with Unit propagation= Generalized arc-consistency Only enclosed area will be explored with unit-propagation 83

79 GSAT local search for SAT (Selman, Levesque and Mitchell, 1992) 1. For i=1 to MaxTries 2. Select a random assignment A 3. For j=1 to MaxFlips 4. if A satisfies all constraint, return A 5. else flip a variable to maximize the score 6. (number of satisfied constraints; if no variable 7. assignment increases the score, flip at random) 8. end 9. end Greatly improves hill-climbing by adding restarts and sideway moves 84

80 With probability p WalkSAT (Selman, Kautz and Cohen, 1994) Adds random walk to GSAT: random walk flip a variable in some unsatisfied constraint With probability 1-p perform a hill-climbing step Randomized hill-climbing often solves large and hard satisfiable problems 85

81 More Stochastic Search Simulated annealing: A method for overcoming local minimas Allows bad moves with some probability: With some probability related to a temperature parameter T the next move is picked randomly. Theoretically, with a slow enough cooling schedule, this algorithm will find the optimal solution. But so will searching randomly. Breakout method (Morris, 1990): adjust the weights of the violated constraints 86